Announcements. The continuity equation Since the fluid is incompressible, the fluid flows faster in the narrow portions of the pipe.

Transcription

1 nnouncements Exam reakdown on Lectures link Exam Wednesday July 8. Last name -K McCC 00, L-Z CSE 0 Reviews Sunday 7:00-9:00, Monday 5:30-7:30, Tuesday 5:30-7:00 N 00 Finish Chapter 9 today Last time we introduced the physics of fluids. This is a very complicated subject. Entire books are dedicated to this topic. The mathematics can be very difficult. We talked about basic physical definitions and we dealt with fluids at rest. Today we talk about fluids in motion. Fluid Flow fluid moving past a surface can exert a viscous force against the surface. This is similar to the frictional force of an object sliding over a surface. We will start by assuming the viscous force to be small. When flow is steady, the velocity at any point is constant in time. The flow may not be the same everywhere. Steady flow is laminar. The streamlines are clearly defined. s we have done many times this semester, we assume the ideal case first. n ideal fluid is incompressible, undergoes laminar flow, and has no viscosity. The continuity equation Since the fluid is incompressible, the fluid flows faster in the narrow portions of the pipe.

2 The mass flow rate is defined as m ρ v The volume flow rate is v The continuity equation for an incompressible fluid equates the volume flow rates past two different points, v v The continuity equation is a consequence of conservation of mass. ernoulli s Equation Using energy ideas, the pressure of the fluid in a constriction cannot be the same as the pressure before or after the constriction. For horizontal flow the speed is higher where the pressure is lower. This is called the ernoulli effect. For a more general situation where the pipe is not horizontal, we can use energy considerations to derive ernoulli s equation. (The derivation is given on page 335. I will only quote the result.) or Hopefully, this reminds you of + ρ gy ρ gy + constant + mv mgy mv W nc + mgy +

3 ernoulli s equation relates the pressure, flow speed, and height at two points in an ideal fluid. roblem 50. Suppose air, with a density of.9 kg/m 3 is flowing into a Venturi meter. The narrow section of the pipe at point has a diameter that is /3 of the diameter of the larger section of the pipe at point. The U-shaped tube is filled with water and the difference in height between the two sections of pipe is.75 cm. How fast is the air moving at point? Strategy Use the continuity equation to relate the speeds at points and. Then, use ernoulli s equation tofind the speed of the air at point. Solution Relate the air speeds at points and.

4 d ( 3) d v v, so v v v 9 v. Find the speed of the air at point. Note that y y ρ(9v ) The pressure difference is related to the height difference in the manometer ρ gh W Subsituting 0 ρ gh W v ρw gh 0ρ 3 (000 kg/m )(9.8 m/s )( (.9 kg/m ) cm).8 m/s roblem 6. In a tornado or hurricane, a roof may tear away from the house because of a difference in pressure between the air inside and the air outside. Suppose that air is blowing across the top of a 000 ft roof at 50 mph. What is the magnitude of the force on the roof? Strategy Use ernoulli s equation to find the pressure difference at the roof. Solution Let the region above the roof be labeled. ssume the air under the roof is still.

5 + + + Now y is almost equal to y and we can assume that the difference in height has negligible effect on the pressure. + Which side is at the higher pressure, the inside or outside? The magnitude of the force on the roof is F ρ airv 3 50 mi h 609 m m (.0 kg m ) (000 ft ) N. h 3600 s mi 3.8 ft which is equal to 56 tons! Viscosity ernoulli s equation ignores viscosity. When real fluids flow, the different layers of fluid drag against each other. pressure difference is needed to maintain the flow. This is similar to needed a constant force to overcome kinetic friction. Fluid layers further away from the wall flow faster than those close to the wall.

6 oiseuille s Law The volume flow rate of viscous fluid through a horizontal cylindrical pipe depends on ressure gradient L Viscosity. The higher the viscosity, the lower the flow rate Radius of the pipe. The French physician oiseuille (pwahzoy) formulated his law after studying blood flow π / L r 8 η Viscosity is η, measured in a-s. Other units are poise (pwaz) and c. roblem. Water flows through a pipe of radius 8.50 mm. The viscosity of water is.005 c. If the flow speed at the center is 0.00 m/s and the flow is laminar, find the pressure drop due to viscosity along a 3.00 m section of pipe. Strategy Use oiseuille s Law to find the pressure drop. Solution oiseuille s law is π / L r 8 η Solving for π / L r 8 η 8 ηl π r The volume flow rate is related to the area of the tube and the speed of the flow (see the continuity equation) v πr v π ( m) (0.00 m/s) m /s Viscosity is not in the correct units. η a-s. The pressure difference is

7 3 8 ηl ( a s)(3 m) (.5 0 m /s) 3 π r π ( m) 67 a problem like this will be on the Exam. Turbulence Unsteady fluid flow. Why do golf balls have dimples? From HowStuffWorks.com: The reason why golf balls have dimples is a story of natural selection. Originally, golf balls were smooth; but golfers noticed that older balls that were beat up with nicks, bumps and slices in the cover seemed to fly farther. Golfers, being golfers, naturally gravitate toward anything that gives them an advantage on the golf course, so old, beat-up balls became standard issue. t some point, an aerodynamicist must have looked at this problem and realized that the nicks and cuts were acting as turbulators they induce turbulence in the layer of air next to the ball (the boundary layer ). In some situations, a turbulent boundary layer reduces drag. If you want to get deeper into the aerodynamics, there are two types of flow around an object: laminar and turbulent. Laminar flow has less drag, but it is also prone to a phenomenon called separation. Once separation of a laminar boundary layer occurs, drag rises dramatically because of eddies that form in the gap. Turbulent flow has more drag initially but also better adhesion, and therefore is less prone to separation. Therefore, if the shape of an object is such that separation occurs easily, it is better to turbulate the boundary layer (at the slight cost of increased drag) in order to increase adhesion and reduce eddies (which means a significant reduction in drag). Dimples on golf balls turbulate the boundary layer. The dimples on a golf ball are simply a formal, symmetrical way of creating the same turbulence in the boundary layer that nicks and cuts do. Viscous Drag n object moving through a fluid experiences drag. Clearly the drag depends on the viscosity of the fluid, the speed of the object, and its size. When the viscous drag is equal to a falling object s weight, the object reaches terminal velocity. This is how parachutes work. Surface Tension The surface of a liquid has special properties not associated with the interior of the liquid. The surface acts like a stretched membrane under tension.